Sparse General Wigner-type Matrices: Local Law and Eigenvector Delocalization

TL;DR

This paper establishes a local law and eigenvector delocalization results for general Wigner-type matrices, including sparse cases, with applications to the Stochastic Block Model and unbounded class scenarios.

Contribution

It introduces new methods that achieve optimal interval length and eigenvector delocalization for dense and sparse Wigner-type matrices, extending previous results to sparser regimes.

Findings

Proved local law for Wigner-type matrices in dense and sparse regimes.

Established eigenvector delocalization for these matrices.

Extended results to the Stochastic Block Model with many classes.

Abstract

We prove a local law and eigenvector delocalization for general Wigner-type matrices. Our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case, and the first results of such kind for the sparse case down to $p=\frac{g(n)\log n}{n}$ with $g(n)\to\infty$. We specialize our results to the case of the Stochastic Block Model, and we also obtain a local law for the case when the number of classes is unbounded.